levels compared to the macroscale, and this, as we’ve seen, has consequences on the electronic and optical properties of the material.
4.3.2 The two- and three-dimensional models
and the concept of degeneracy
The particle on a line model can be extended to describe the particle
moving in both a two-dimensional rectangular region (a quantum well)
and a three-dimensional cubical region (a quantum cube). Let’s first
discuss the two-dimensional model. Figure 4.10a shows a rectangle of
length a and b along the coordinates x and y, respectively. In order to
solve the Schrödinger equation for this system, we require the twodimensional kinetic energy operator described by Equation 4.21:
^
H = −
h
2
8π 2 m
∂
2
∂x 2 +
∂
2
∂y 2
(4.21)
The wavefunction we need now depends on the two variables x and y.
One way of describing such a wavefunction is to write it as a product of two
functions, one that depends only on x and one that depends only on y.
Splitting up the wavefunction in this way is known as a separation of
variables and is commonly used to describe wavefunctions that depend
on more than one variable. In this case we can write
y x, y
ð Þ = X x
ð ÞY y
ð Þ
(4.22)
The two-dimensional wavefunction can be reduced to the product of two
one-dimensional wavefunctions, each of which is described by Equation
4.14. Thus
y x, y
ð Þ =
ffiffiffi ffi
2
a
r
sin
n x πx
a
ffiffiffi ffi
2
b
r
sin
n y πy
b
=
ffiffiffiffiffiffi
4
ab
r
sin
n x πx
a
sin
n y πy
b
(4.23)
b
(a)
(b)
y
b
y
a
a
x
x
z
c
V(x,y) = 0
V(x,y,z) = 0
Figure
4.10 The
twodimensional (a) and threedimensional
(b)
regions
within which the particle
resides. The wavefunction
and potential energy outside
of the region is zero and
infinite, respectively.
CONFINEMENT OF ELECTRONS IN BOXES 113
4.3.2 The two- and three-dimensional models
and the concept of degeneracy
The particle on a line model can be extended to describe the particle
moving in both a two-dimensional rectangular region (a quantum well)
and a three-dimensional cubical region (a quantum cube). Let’s first
discuss the two-dimensional model. Figure 4.10a shows a rectangle of
length a and b along the coordinates x and y, respectively. In order to
solve the Schrödinger equation for this system, we require the twodimensional kinetic energy operator described by Equation 4.21:
^
H = −
h
2
8π 2 m
∂
2
∂x 2 +
∂
2
∂y 2
(4.21)
The wavefunction we need now depends on the two variables x and y.
One way of describing such a wavefunction is to write it as a product of two
functions, one that depends only on x and one that depends only on y.
Splitting up the wavefunction in this way is known as a separation of
variables and is commonly used to describe wavefunctions that depend
on more than one variable. In this case we can write
y x, y
ð Þ = X x
ð ÞY y
ð Þ
(4.22)
The two-dimensional wavefunction can be reduced to the product of two
one-dimensional wavefunctions, each of which is described by Equation
4.14. Thus
y x, y
ð Þ =
ffiffiffi ffi
2
a
r
sin
n x πx
a
ffiffiffi ffi
2
b
r
sin
n y πy
b
=
ffiffiffiffiffiffi
4
ab
r
sin
n x πx
a
sin
n y πy
b
(4.23)
b
(a)
(b)
y
b
y
a
a
x
x
z
c
V(x,y) = 0
V(x,y,z) = 0
Figure
4.10 The
twodimensional (a) and threedimensional
(b)
regions
within which the particle
resides. The wavefunction
and potential energy outside
of the region is zero and
infinite, respectively.
CONFINEMENT OF ELECTRONS IN BOXES 113